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Properties

Label 2-60-4.3-c6-0-22
Degree $2$
Conductor $60$
Sign $-0.988 + 0.149i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.21 − 6.06i)2-s − 15.5i·3-s + (−9.54 − 63.2i)4-s + 55.9·5-s + (−94.5 − 81.3i)6-s − 234. i·7-s + (−433. − 272. i)8-s − 243·9-s + (291. − 338. i)10-s − 611. i·11-s + (−986. + 148. i)12-s + 446.·13-s + (−1.41e3 − 1.22e3i)14-s − 871. i·15-s + (−3.91e3 + 1.20e3i)16-s − 4.75e3·17-s + ⋯
L(s)  = 1  + (0.652 − 0.758i)2-s − 0.577i·3-s + (−0.149 − 0.988i)4-s + 0.447·5-s + (−0.437 − 0.376i)6-s − 0.682i·7-s + (−0.846 − 0.531i)8-s − 0.333·9-s + (0.291 − 0.338i)10-s − 0.459i·11-s + (−0.570 + 0.0861i)12-s + 0.203·13-s + (−0.517 − 0.445i)14-s − 0.258i·15-s + (−0.955 + 0.294i)16-s − 0.967·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.988 + 0.149i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.988 + 0.149i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.165320 - 2.20422i\)
\(L(\frac12)\) \(\approx\) \(0.165320 - 2.20422i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.21 + 6.06i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 + 234. iT - 1.17e5T^{2} \)
11 \( 1 + 611. iT - 1.77e6T^{2} \)
13 \( 1 - 446.T + 4.82e6T^{2} \)
17 \( 1 + 4.75e3T + 2.41e7T^{2} \)
19 \( 1 - 4.22e3iT - 4.70e7T^{2} \)
23 \( 1 + 3.89e3iT - 1.48e8T^{2} \)
29 \( 1 - 1.52e4T + 5.94e8T^{2} \)
31 \( 1 + 2.57e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.29e4T + 2.56e9T^{2} \)
41 \( 1 - 7.11e4T + 4.75e9T^{2} \)
43 \( 1 + 1.01e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.32e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.83e5T + 2.21e10T^{2} \)
59 \( 1 - 1.76e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.19e5T + 5.15e10T^{2} \)
67 \( 1 + 4.20e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.27e5iT - 1.28e11T^{2} \)
73 \( 1 - 2.76e5T + 1.51e11T^{2} \)
79 \( 1 + 6.16e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.32e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.04e6T + 4.96e11T^{2} \)
97 \( 1 + 1.31e6T + 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.43409491686682332963220849085, −12.29981990421313043109011752134, −11.12440713037672574803649534935, −10.15475324398821524153700862546, −8.699947639655898879032866757140, −6.87206269139128848656954223373, −5.66024602452630111479177993184, −3.98978563590489744839047808519, −2.28473233806204523917915933749, −0.73948307431175904777220097262, 2.66086829439602426057537536257, 4.40715092992226145348860737343, 5.58372169696462110372417238869, 6.83218344140955503182534520870, 8.488714596729372255883543953457, 9.464266943334699264088792963217, 11.10975005441676522611011374303, 12.36555651672082315375480611110, 13.44305736327452890659294092610, 14.52677297908383949851281721896

Graph of the $Z$-function along the critical line